Maximal k-independent sets in graphs

Mostafa Blidia; Mustapha Chellali; Odile Favaron; Nacéra Meddah

Displaying similar documents to “Maximal k-independent sets in graphs”

On characterization of uniquely 3-list colorable complete multipartite graphs

Yancai Zhao, Erfang Shan (2010)

Discussiones Mathematicae Graph Theory

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For each vertex v of a graph G, if there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k-list colorable graph. Ghebleh and Mahmoodian characterized uniquely 3-list colorable complete multipartite graphs except for nine graphs: $K_{2, 2, r}$ r ∈ 4,5,6,7,8, $K_{2, 3, 4}$ , $K_{1 * 4, 4}$ , $K_{1 * 4, 5}$ , $K_{1 * 5, 4}$ . Also, they conjectured that the nine graphs are not U3LC graphs. After that, except for $K_{2, 2, r}$ r ∈ 4,5,6,7,8, the others have been proved not...

On the total k-domination number of graphs

Adel P. Kazemi (2012)

Discussiones Mathematicae Graph Theory

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Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number $γ_{\times k} (G)$ of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, $| N_{G} [v] \cap S | \geq k$ . Also the total k-domination number $γ_{\times k, t} (G)$ of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, $| N_{G} (v) \cap S | \geq k$ . The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H). We know that for...

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

Paired domination in prisms of graphs

Christina M. Mynhardt, Mark Schurch (2011)

Discussiones Mathematicae Graph Theory

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The paired domination number $γ_{p r} (G)$ of a graph G is the smallest cardinality of a dominating set S of G such that ⟨S⟩ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: $γ_{p r} (π G) = 2 γ_{p r} (G)$ for all πG; $γ_{p r} (K ₂ ☐ G) = 2 γ_{p r} (G)$ ; $γ_{p r} (K ₂ ☐ G) = γ_{p r} (G)$ .

Generalized connectivity of some total graphs

Yinkui Li, Yaping Mao, Zhao Wang, Zongtian Wei (2021)

Czechoslovak Mathematical Journal

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We study the generalized $k$ -connectivity $κ_{k} (G)$ as introduced by Hager in 1985, as well as the more recently introduced generalized $k$ -edge-connectivity $λ_{k} (G)$ . We determine the exact value of $κ_{k} (G)$ and $λ_{k} (G)$ for the line graphs and total graphs of trees, unicyclic graphs, and also for complete graphs for the case $k = 3$ .

Roman bondage in graphs

Nader Jafari Rad, Lutz Volkmann (2011)

Discussiones Mathematicae Graph Theory

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A Roman dominating function on a graph G is a function f:V(G) → 0,1,2 satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value $f (V (G)) = \sum_{u \in V (G)} f (u)$ . The Roman domination number, $γ_{R} (G)$ , of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage $b_{R} (G)$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E’ ⊆ E(G)...

The extremal irregularity of connected graphs with given number of pendant vertices

Xiaoqian Liu, Xiaodan Chen, Junli Hu, Qiuyun Zhu (2022)

Czechoslovak Mathematical Journal

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The irregularity of a graph $G = (V, E)$ is defined as the sum of imbalances $| d_{u} - d_{v} |$ over all edges $u v \in E$ , where $d_{u}$ denotes the degree of the vertex $u$ in $G$ . This graph invariant, introduced by Albertson in 1997, is a measure of the defect of regularity of a graph. In this paper, we completely determine the extremal values of the irregularity of connected graphs with $n$ vertices and $p$ pendant vertices ( $1 \leq p \leq n - 1$ ), and characterize the corresponding extremal graphs.

Extremal properties of distance-based graph invariants for $k$ -trees

Minjie Zhang, Shuchao Li (2018)

Mathematica Bohemica

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Sharp bounds on some distance-based graph invariants of $n$ -vertex $k$ -trees are established in a unified approach, which may be viewed as the weighted Wiener index or weighted Harary index. The main techniques used in this paper are graph transformations and mathematical induction. Our results demonstrate that among $k$ -trees with $n$ vertices the extremal graphs with the maximal and the second maximal reciprocal sum-degree distance are coincident with graphs having the maximal and the second...

On choosability of complete multipartite graphs $K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}$

Guo-Ping Zheng, Yu-Fa Shen, Zuo-Li Chen, Jin-Feng Lv (2010)

Discussiones Mathematicae Graph Theory

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A graph G is said to be chromatic-choosable if ch(G) = χ(G). Ohba has conjectured that every graph G with 2χ(G)+1 or fewer vertices is chromatic-choosable. It is clear that Ohba’s conjecture is true if and only if it is true for complete multipartite graphs. In this paper we show that Ohba’s conjecture is true for complete multipartite graphs $K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}$ for all integers t ≥ 1 and k ≥ 2t+2, that is, $c h (K_{4, 3 * t, 2 * (k - 2 t - 2), 1 * (t + 1)}) = k$ , which extends the results $c h (K_{4, 3, 2 * (k - 4), 1 * 2}) = k$ given by Shen et al. (Discrete Math. 308 (2008) 136-143), and $c h (K_{4, 3 * 2, 2 * (k - 6), 1 * 3}) = k$ ...

Remarks on $D$ -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

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A graph is called distance integral (or $D$ -integral) if all eigenvalues of its distance matrix are integers. In their study of $D$ -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$ -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1}, p_{2}, p_{3}}$ with $p_{1} < p_{2} < p_{3}$ , and $K_{p_{1}, p_{2}, p_{3}, p_{4}}$ with $p_{1} < p_{2} < p_{3} < p_{4}$ , as well as the infinite classes of distance integral...

Domination and independence subdivision numbers of graphs

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi (2000)

Discussiones Mathematicae Graph Theory

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The domination subdivision number $s d_{γ} (G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of...

A note on the independent domination number versus the domination number in bipartite graphs

Shaohui Wang, Bing Wei (2017)

Czechoslovak Mathematical Journal

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Let $γ (G)$ and $i (G)$ be the domination number and the independent domination number of $G$ , respectively. Rad and Volkmann posted a conjecture that $i (G) / γ (G) \leq Δ (G) / 2$ for any graph $G$ , where $Δ (G)$ is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than $Δ (G) / 2$ are provided as well.

Nearly complete graphs decomposable into large induced matchings and their applications

Noga Alon, Ankur Moitra, Benjamin Sudakov (2013)

Journal of the European Mathematical Society

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We describe two constructions of (very) dense graphs which are edge disjoint unions of large induced matchings. The first construction exhibits graphs on $N$ vertices with $(\frac{N}{2}) - o (N^{2})$ edges, which can be decomposed into pairwise disjoint induced matchings, each of size $N^{1 - o (1)}$ . The second construction provides a covering of all edges of the complete graph $K_{N}$ by two graphs, each being the edge disjoint union of at most $N^{2 - δ}$ induced matchings, where $δ > 0, 076$ . This disproves (in a strong form) a conjecture of Meshulam,...

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

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In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g (n, l)$ , the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$ . By edit distance of two graphs $G$ , $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$ . This new extremal number $g (n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...